Confusing identities in lambert w function that W(x)e^W(x)=x=W(xe^x), how is this possible ?
You’ve hit on the most important — and initially confusing — property of the Lambert W function. Let’s break it down step by step.
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1. Definition of \( W \)
The Lambert \( W \) function is defined as the inverse of the function:
\[
f(w) = w e^w
\]
That means:
\[
\text{If } y = w e^w, \quad \text{then } w = W(y).
\]
So by definition :
\[
W(x) e^{W(x)} = x
\]
for any \( x \) where \( W(x) \) is defined.
—
2. Second identity: \( W(x e^x) = x \)
Let’s check this one carefully.
We want to solve for \( W(x e^x) \).
Let \( t = x e^x \).
By definition of \( W \), \( W(t) \) is the number such that:
\[
W(t) e^{W(t)} = t
\]
But \( t = x e^x \), so:
\[
W(x e^x) e^{W(x e^x)} = x e^x
\]
Now, notice that \( x \) itself satisfies \( x e^x = t \).
Since \( W \) is the inverse of \( w \mapsto w e^w \), applying \( W \) to \( t \) should return \( x \) if \( x \) is in the correct domain (i.e., \( x \ge -1 \) for the principal branch \( W_0 \)).
So indeed:
\[
W(x e^x) = x
\]
for \( x \ge -1 \) (principal branch).
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3. Why both identities look similar but are different
First identity: \( W(x) e^{W(x)} = x \)
This is the direct definition of \( W \): “\( W(x) \) is the number whose product with \( e^{W(x)} \) equals \( x \).”
Second identity: \( W(x e^x) = x \)
This is the inverse property: “If you take \( x \), form \( x e^x \), then apply \( W \), you get back \( x \).”
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4. Analogy with logarithm
Think of the natural log function \( \ln(x) \):
– Definitional property: \( e^{\ln x} = x \)
– Inverse property: \( \ln(e^x) = x \)
Similarly for \( W \):
– Definitional: \( W(x) e^{W(x)} = x \)
– Inverse: \( W(x e^x) = x \)
—
5. Example with numbers
Let \( x = 2 \):
– Compute \( W(2 e^2) \):
\( 2 e^2 \approx 14.778 \)
\( W(14.778) \) is the number \( w \) such that \( w e^w = 14.778 \).
Try \( w=2 \): \( 2 e^2 \approx 14.778 \) ✓
So \( W(2 e^2) = 2 \). This illustrates \( W(x e^x) = x \).
– Compute \( W(2) \):
\( W(2) \approx 0.8526 \)
Check: \( 0.8526 \times e^{0.8526} \approx 0.8526 \times 2.345 = 2.000 \) ✓
This illustrates \( W(x) e^{W(x)} = x \).
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6. Domain considerations
– \( W(x e^x) = x \) holds for \( x \ge -1 \) for the principal branch \( W_0 \), and for \( x \le -1 \) for the \( W_{-1} \) branch.
– \( W(x) e^{W(x)} = x \) holds for all \( x \) in the domain of \( W \) (which is \( x \ge -1/e \) for real \( W \)).
—
7. Summary
Both identities express that \( W \) and the map \( w \mapsto w e^w \) are inverses:
– \( W(x) e^{W(x)} = x \) says: applying \( w e^w \) to \( W(x) \) gives \( x \).
– \( W(x e^x) = x \) says: applying \( W \) to \( x e^x \) gives \( x \).
They are two sides of the same coin — one is “function inverse in one direction,” the other is “function inverse in the other direction.”